<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K15" title="GLIB_004:func.15">PRIM:MST</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> </span><br/><b>let </b><font color="Maroon" title="c2">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K15" title="GLIB_004:func.15">PRIM:MST</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> </span><br/>



<b>set </b><font color="Maroon" title="c3">PCS</font> =  <a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b><span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K15" title="GLIB_004:func.15">PRIM:MST</a> <font color="Maroon" title="c1">G</font>;<br/>
<a NAME="E1:86"/><i><font color="Green" title="E56">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<div><i><font color="Green" title="E57">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><b>assume </b><a NAME="E1:86_1"/><i><font color="Green" title="E58">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><b>set </b><font color="Maroon" title="c5">off</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> ;<br/><b>set </b><font color="Maroon" title="c6">Gn</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c7">Gn1</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c8">Next</font> =  <a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon" title="c9">e</font> =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p1">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span>;<br/><a NAME="E2:86_1"/><i><font color="Green" title="E59">A4</font></i>: 
<span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_004.html#K13" title="GLIB_004:func.13">PRIM:Step</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_004.html#D15" target="_self" title="GLIB_004:def.15">Def15</a></i>;<br/><a NAME="E3:86_1"/><i><font color="Green" title="E60">A5</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E1:86_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="txt" href="glib_004.html#E68"><i><font color="Green" title="E52">Lm18</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:86_1_1"/>
 <a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c10">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:86_1_1"/><i><font color="Green" title="E61">A6</font></i>: 
( not <font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="glib_003.html#K14" title="GLIB_003:func.14">.labeledV()</a>  &amp; <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span> <a href="glib_003.html#K12" title="GLIB_003:func.12">.labelEdge</a> <span class="p2">(<span class="default"><a href="subset_1.html#K10" title="SUBSET_1:func.10">choose</a> <span class="p3">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">PRIM:NextBestEdges</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span></span>)</span></span>)</span>,1</span>)</span> <a href="glib_003.html#K13" title="GLIB_003:func.13">.labelVertex</a> <font color="Maroon" title="c10">v</font>,1 )
 <b>by </b><i><a class="txt" href="glib_004.html#E2:86_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="txt" href="glib_004.html#E56"><i><font color="Green" title="E40">Lm6</font></i></a></i>;<br/><a NAME="E4:86_1_1"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="glib_004.html#E58"><i><font color="Green" title="E42">Lm8</font></i></a></i>;<br/><a NAME="E5:86_1_1"/><b>then </b>
 <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/><b>hence </b><a NAME="E6:86_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_004.html#E3:86_1"><i><font color="Green" title="E60">A5</font></i></a>, <a class="txt" href="glib_004.html#E3:86_1_1"><i><font color="Green" title="E61">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E5:86_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="glib_004.html#E1:86_1"><i><font color="Green" title="E58">A3</font></i></a>, <a class="txt" href="glib_004.html#E2:86_1"><i><font color="Green" title="E59">A4</font></i></a>, <a class="ref" href="glib_004.html#D14" target="_self" title="GLIB_004:def.14">Def14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E3:86"/><i><font color="Green" title="E58">A7</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="glib_004.html#E1:86"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="glib_004.html#E2:86"><i><font color="Green" title="E57">A2</font></i></a>);<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:86_2_1"/>
( <font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a>  or <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_1_1"><b>suppose </b></a><a NAME="E1:86_2_1_1"/>
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K15" title="GLIB_004:func.15">PRIM:MST</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> </span><br/><div class="add"><b>hence </b><a NAME="E2:86_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K15" title="GLIB_004:func.15">PRIM:MST</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E63"><i><font color="Green" title="E47">Lm13</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="86_2_1_2"><b>suppose </b></a><a NAME="E1:86_2_1_2"/>
<span class="p1">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K15" title="GLIB_004:func.15">PRIM:MST</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> </span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c4">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E3:86_2_1_2"/><i><font color="Green" title="E59">A8</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c4">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><b>thus </b><a NAME="E4:86_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K15" title="GLIB_004:func.15">PRIM:MST</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> 
 <b>by </b><i><a class="txt" href="glib_004.html#E3:86"><i><font color="Green" title="E58">A7</font></i></a>, <a class="txt" href="glib_004.html#E3:86_2_1_2"><i><font color="Green" title="E59">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:86"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K14" title="GLIB_004:func.14">PRIM:CompSeq</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">.</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="glib_004.html#K15" title="GLIB_004:func.15">PRIM:MST</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="glib_003.html#K11" title="GLIB_003:func.11">.labeledE()</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
